Analysis of Onset and Development of Ore Formation in Dajishan Tungsten Ore Area, Jiangxi Province, China

INTRODUCTION

Geological systems are mostly non-linear systems, and the metallogenic systems are in particular strongly non-linear systems. Under the strong nonlinear conditions, the dynamics is bound to develop toward complexity (Yu, 2006). Hence, the geological and metallogenic systems belong to a class of complex dynamic systems. A number of scholars at home and abroad have applied non-linear sciences and theory of complexity to studying various problems of complexity in geosystems, such as fractal and multifractal, dissipative structures and self-organized criticality. Ortoleva (1994) systematically studied the phenomena of self-organization in geochemistry; Hergarten (2002) discussed self-organized criticality in detail in the earth system. Sornette(2004, 2000) published a monograph on critical phenomena in natural sciences, etc..

In recent years, Yu(2006, 2003) applied nonlinear sciences and the theory of complexity in studying geosystems and metallogenic systems and published the corresponding monographs of "the complexity of geosystems", and "fractal growth of mineral deposits at the edge of chaos" is a new theory of genesis of mineral deposits. He has succeeded in applying it in researches on genesis of many kinds of ore deposits and the regularity of regional distribution of the mineral resources in Nanling metallogenic province and the district around the rim of the Yangtze craton in South China.

Sornette put forward new notions of "complex dimension" and "discrete-scale invariance" in 1998, and advanced the new "theory of self-similar oscillatory finite-time singularities" in 2002, 2003. He applied the theory in the prediction of many catastrophic events and acquired satisfactory effects (Sornette, 2003, 2002, 1998).

In this study, the authors want to use the geochemical data (scale of 1/50 000) of dispersion flow accomplished in the year 2000 for the Dajishan ore area, and try to apply the theory to studying and analyzing the onset and development of ore formation in the Dajishan tungsten ore area, Jiangxi Province, China.

GEOLOGICAL BACKGROUND OF THE STUDIED AREA

The Dajishan tungsten ore area of Jiangxi Province is situated in the southeastern part of the Nanling region in South China. Nanling region is the district with the most abundant mineral resource of tungsten in China and also in the world.

The regional basement of the studied area consists of strata of the Sinian and Cambrian, and is covered by strata of the Devonian, Carboniferous, Triassic, Jurassic and Cretaceous. The regional lines of structure are primarily in the directions of northeast and east-west, and the north-northeast and northwest directed structures are of the second order. The IndoChina movement and Yanshan movement are the strongest orogeny in the district, and the magmatic intrusion was very active.

The Dajishan tungsten mine is a high temperature hydrothermal ore deposit of the wolframite-quartz vein type. It has the characteristics of the so-called "five-storeyed" vertical zonality. The region's geological structure is located at the transitional part between the Wuyishan uplift and the Yuebei depression. The ore area is situated in the eastern part beyond the east-west spreading granite belt of GuposhanDadongshan-Guidong. The igneous rocks related to the ore formation are two-mica granite and muscovite granite.

DYNAMIC MODEL

Singularities play an important role in the physics of phase transition as well as in signatures of positive feedbacks in dynamic systems. The mathematics of singularities is applied routinely in the physics of phase transitions to describe for instance the transformations from ice to water or from a magnet to a demagnetized state when raising the temperature, as well as in many other condensed matter systems. Such singularities characterize the so-called critical phenomena. Other classes of singularities occur in dynamic systems and are spontaneously reached in finite time. Spontaneous singularities in ordinary differential equation (ODE) and partial differential equation (PDE) are quite common and have been found in many well-established models of natural systems, such as in the Euler equations of inviscid fluids, models of rupture and material failure, earthquakes, stock market crashes, population dynamics and human parturition (Sornette and Ide, 2003; Ide and Sornette, 2002; Johansen and Sornette, 2000).

During the third century BC, Euclid and his students introduced the concept of space dimension, which can take positive integer values equal to the number of independent directions. We had to wait until the second half of the 19th century and the 20th century to witness the generalization of dimensions to fractional values. The word "fractal" was coined by Mandelbrot to describe sets consisting of parts similar to the whole, and which can be described by a fractional dimension. This generalization of the notion of a dimension from integers to real numbers reflects the conceptual jump from translational invariance to continuous scale invariance (Sornette, 1998).

Sornette (1998) further generalized the concept of fractal dimension from real to "complex number" and extended "continuous scale invariance" to "discrete scale invariance". "Discrete scale invariance" is a weaker form of scale-invariance symmetry, associated with log-periodic corrections to scaling. The observable signature of discrete scale invariance is the presence of log-periodic oscillations.

Sornette and Ide (2003), Sornette (2002) had carried out systematic in-depth researches on many kinds of "catastrophic events" (for example, materials rupture, earthquakes, turbulence, climate catastrophe, financial crash, human parturition and population dynamics, etc.) from complex systems in nature to the human society. They found that the sudden transition from stasis to crisis of a catastrophic event conforms to the following universal law. It occurs by way of discrete scale invariance followed by continuous scale invariance to arrive at the singularities.

Then, Sornette and Ide (2003), Sornette (2002) put forward the "theory of self-similar oscillatory finite-time singularities". It states: a two-dimensional dynamic system with two non-linear terms exerting respectively positive feedback and negative feedback, by competing with each other, reaches the singularities in finite time by virtue of self-similar log-periodic oscillations (discrete-scale invariance) and succeeded by power-law accelerating growth. They applied his theory to predicting catastrophic events of complex systems and achieved good results.

The dynamic equations are

where y₁ and y₂ are two non-linear terms, y₂ for positive feedback, y₁ for negative feedback; α is the measure for the effect of positive feedback; γ is the measure for the effect of negative feedback; m > 1, n > 1; | | is the symbol for absolute value.

Putting the non-linear oscillation and positive feedback terms together, Sornette and Ide (2003), Sornette (2002) analyzed the overall dynamics of equations (1) and (2) which are characterized in Fig. 1. The two special intertwined trajectories along b⁺ (solid line) and b^- (dashed line) connect the origin (0, 0) to (+∞, +∞) and (-∞, -∞), respectively, and hence divide the phase space y≡(y₁, y₂) into two distinct basins B⁺ and B^-. The basin B⁺ (resp. B^-) corresponds to a finite-time singularity y_c(y₀) with y_2c(y₀)=+∞ (resp. y_2c(y₀)=-∞) but finite y_1c(y₀) at the critical time t_c(y₀). Starting from y₀ in B⁺ close to the origin at t₀, a trajectory y (t; y₀, t₀) spirals out with clockwise rotation and we count a turn each time it crosses the y₁-axis, i.e., y₁ changes its direction of motion (dy₁/dt=0). Deep in the spiral structure, y(t; y₀, t₀) follows approximately the orbit of constant Hamiltonian H defined by the non-linear oscillator but fails to close on itself because H is no longer conserved due to the positive feedback. This slowly growing non-linear oscillator defines the first dynamic regime (oscillatory behavior). Any trajectory starting in the first dynamic regime must eventually cross-over to the second one (singular behavior) associated with a route to the singularity without any further oscillation. Figure 2 shows a typical time series of a y(t; y₀, t₀) starting from y₀ near the origin. In the second dynamic regime, y₂ diverges (and therefore dy₂/dt also diverges) while y₁ remains finite on the approach to t_c(y₀). As a consequence, the reversal term γy₁|y₁|^n–1 in dy₂/dt, equation (2), becomes negligible close to t_c(y₀) (Sornette and Ide, 2003).

Figure  1.  Geometry of the boundaries b⁺ and b^- as well as the basins B⁺ and B^- for (n, m)=(3, 2.5) and γ=10 in phase space. The exit segments Δe^{(+1, +0)}∈ B⁺ and Δe^{(-1, +0)}∈B^- are thick solid and dashed lines respectively on the y₁-axis; the squares and diamonds are the initial condition and exit point of the trajectory in Fig. 2. The exit point is the point of transition from oscillatory to singular behavior (Sornette and Ide, 2003).

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Figure  2.  A trajectory in B⁺ starting from initial condition (-2.328 78×10^-2, 1.710 83×10^-2) at t=-500, going through the exit point (0.5, 0) at t=0, to the critical point (0.768 899, +∞) at t_c=2.672 84. Solid and dashed lines correspond to y₁ and y₂, respectively; diamond and triangle are the points in the exit segment Δe^{(+1, +0)} (i.e., the point of transition from oscillatory to singular behavior), and the critical point along the trajectory with open and filled symbols corresponding to y₁ and y₂ (Sornette and Ide, 2003).

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The authors try to apply this theory to studying and analyzing the onset and development of ore formation in the Dajishan tungsten ore area, Jiangxi Province. The formation of the Dajishan tungsten ore deposit is a complex dynamic process. However, the dissolution-precipitation reaction is the dominant process. This dynamic system consists of two processes: one is the precipitation by way of positive feedback of the activator, and the other is the dissolution by way of negative feedback of the inhibitor. As for the tungsten ore deposit, the former is the dominant process. That is to say, in the whole ore-forming process, the precipitation owing to positive feedback competes with the dissolution owing to negative feedback, and the former is superior to the latter, therefore, the ore-forming system reaches the singularity in finite time and accomplishes the whole process of ore formation. This process is well in compliance with equations (1) and (2). Thus, it is natural to regard the chemical element tungsten as an activator, take its concentration as the carrier of positive feedback, and apply the "theory of self-similar oscillatory finite-time singularities" to investigating how the element tungsten realized the onset and development of ore formation under the constraint of negative feedback of the inhibitor. In order to study the spatiotemporal evolution of the chemical element tungsten as well as to accurately characterize the signatures of self-similar log-periodic oscillations of the positive and negative feedback (the local extrema), the authors choose the derivative of logarithms of contents of the chemical element tungsten M(r) with respect to that of distance to the origin r as the variable to be studied

where Dr is local fractal dimension.

Studies have shown that discrete scale invariance and its associated complex exponents and logperiodicity may appear "spontaneously" in natural systems (Ide and Sornette, 2002; Sornette, 1998). Dr as a local fractal dimension is quite sensitive to selfsimilar log-periodic oscillations, so it can be used to effectively reveal these oscillations in the complex ore-forming processes.

Then the authors select the center of the Dajishan tungsten ore deposit as the starting point for the ore-forming processes. The identification of the transition point from oscillatory to singular behavior is the crux when we come to use "discrete-scale invariance" to predict the occurrence of some catastrophic events. In dynamics, in order to find the transition point, we have to trace back to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin (Fig. 1). Here, the origin plays a particularly important role. This argument meets the need of researches on the onset and development of ore formation in the present work. It is natural to choose the center of the Dajishan ore deposit as the reference point for prediction.

In order to study and analyze the onset and development of ore formation in the studied ore area, the Dr versus lnr plots are used to illustrate the onedimension trend of evolution of the ore-forming processes.

METHOD AND RESULTS OF RESEARCH

The geochemical field has the dual attributes of stochasticity and structurality. The authors apply the theory of "geostatistics" and method of "Kriging" to reveal the geochemical characteristics of the Dajishan tungsten ore area.

Geostatistics is a theory of the stochastic field that takes spatial correlation (spatio-temporal structure of mineralization) as the basis, the regionalized variable as the nucleus, and the variogram function as the mathematical tool. "Kriging" is a method of local estimation. It gives unbiased estimates of average grades of ores with the least variance of estimation.

The authors used the geochemical data (scale of 1/50 000) of dispersion flow accomplished in 2000 for the Dajishan ore area by the fifth team of Geological Exploration for Nonferrous Metals of Jiangxi Province. The prospect area is 405 km², in which 1 170 sampling sites were allocated. The concentrations of 19 chemical elements (Au, Ag, As, Sb, Cu, Pb, Zn, W, Sn, Mo, Bi, Cr, Ni, Co, V, Ti, Ba, Mn, Be) were analyzed, and 22 230 data in total were provided.

The results of calculation were used to illustrate the distribution of chemical element tungsten (Fig. 3). It shows that the quantitative distribution of the chemical element tungsten is mainly concentrated within an area centered at the Dajishan mine and with a radius of 2 km. Several small tungsten anomalies are localized outside the ore area and along the directions of NNW, SW and SE. Then six representative profiles AO, BO, CO, DO, EO and FO are selected for researches (Fig. 3).

Figure  3.  The quantitative distribution of chemical element tungsten and the locations of profiles in Dajishan ore area.

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Two kinds of diagrams have been completed, i.e., M(r) versus r plot (Fig. 4), and Dr versus lnr plot (Fig. 5). Figure 4 shows that all the six profiles reflect the similar characteristics. During the evolution from outside the Dajishan ore area toward its center, the variation of M(r) undergoes a process of natural and gentle oscillation, succeeded by power-law accelerating growth and then approaches the center of ore formation. The point is that the acceleration occurs at a distance very near the center of ore formation (r=2–3). This is manifested most clearly in the profile AO.

Figure  4.  M(r) versus r plots of the chemical element tungsten in Dajishan ore area.

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Figure  5.  Dr versus lnr plots of the chemical element tungsten in Dajishan ore area.

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Figure 5 shows that the variation of local fractal dimension Dr undergoes a process of self-similar log-periodic oscillations, succeeded by power-law accelerating growth and then approaches the center of ore formation. This fact reflects that at lnr=0–1, the spatio-temporal evolution of the chemical element tungsten passes through the critical transition point, from the non-critical region crosses over to the critical region and arrives at the singularity where the onset of ore formation appears (Fig. 6).

Figure  6.  Diagrammatic illustration of "theory of self-similar oscillatory finite-time singularities" (Yu and Peng, 2009).

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CONCLUSIONS

The authors try to apply the "theory of selfsimilar oscillatory finite-time singularities" to studying and analyzing the onset and development of ore formation in the Dajishan tungsten ore area, Jiangxi Province and draw the following conclusions.

In the Dajishan tungsten ore area, the contents of the chemical element tungsten play the role of carrier of positive feedback (or activator) in the ore-forming processes. The mutual competition of positive and negative feedback gives rise to self-similar logperiodic oscillations succeeded by power-law accelerating growth.

The ore-forming system correspondingly proceeded from the non-critical region, crossed over the critical transition point to enter into the critical region, and eventually approached to singularity in finite time. By this way the onset of ore formation up to its completion was realized (Fig. 6).

This regularity satisfactorily reproduced the dynamic processes of ore-formation of the studied ore deposit. The authors hold that the Sornette's theory can be further applied to the prediction of prospecting areas.